Wave-particle duality#

What you need to know

  • Compton scattering and electron diffraction experimentshave demonstrated that concepts of a particle and wave are not mutually exclusive.

  • A physical entity has both wave-like (wavelengths, interference, diffraction, etc.) and particle-like (momentum, collision, countable, etc.) characteristics.

  • Wave-like and particle-like characteristics are inversely proportional to each other and are quantified by de Broglie relation: \(\lambda = \frac{h}{p}\)

  • An electron has a wavelength; a photon has momentum. Welcome to the reality where everything can behave both as a wave and a particle!

Electron diffraction#

  • Davisson and Germer at Bell Labs showed that electrons exhibit diffraction. Diffraction was previously thought to be classical wave pheonomenon.

  • Electron diffraction and neutron diffraction are widely used experimental techniques today, complementing X-ray diffraction.

Diffraction and color.#

  • Thin films of oil, soap bubbles, butterfly wings, seashells, CD and DVD, etc. show a rainbow pattern. This is caused by the constructive and destructive interference of light traveling slightly different paths.

  • An electron can be scattered by different layers of a crystal lattice and interfere constructively or destructively, giving rise to alternating intensity patterns.

Bragg’s formula for diffraction.#

  • X-rays interact with the atoms in a crystal. The phase shift upon scattering off of atoms causes constructive (left figure) or destructive (right figure) interferences.

  • Maxima and minima in interference patters are dictated by simple geometric argument captured in Bragg’s formula:

\[\boxed{2d sin\theta = n\lambda}\]

  • One expects waves like X-rays to show interference patterns according to this formula. Interference was thought to be a purely a wave like phenomenon.

Both X-rays and electrons show the same diffraction patterns.#

Davisson & Germer show: electrons scatter like waves.#

  • In 1925, Davisson and Germer were studying electron scattering from various materials. To their great surprise, they discovered that at certain angles there was a peak in the intensity of the scattered electron beam. This peak indicated wave behavior for the electrons and could be interpreted by Bragg’s law (previously only applied to X-ray scattering) to give values for the lattice spacing in the nickel crystal.

Particles can behave like waves in appropriate circumstances.#

  • Observing diffraction patterns and computing wavelength from de Broglie relation confirmed that thinking of matter as a dual wave-particle was correct with an impressive agreement between experimental predictions and theory.

  • Another striking wave behavior of electrons emerges in the double-slit experiment where electrons all line up as bands on the detector screen after passing throw a double-slit wall. We are going to have a more in-depth look into this incredible experiment later on.

Compton scattering#

Arthur Compton showed that X-rays get scattered off free electrons like elastic billiard balls. Applying conservation of momentum principle (previously only applied to particle-like objects), it was shown that the outgoing X-rays should be of longer wavelength than the incoming ones. This means that a moving photon hits the resting free electron and transfers some energy to get the electron moving. Note that this experimental result makes sense only if you think of a photon as a particle with linear momentum which gets bounced off the electron.

Wave particle duality as universal feature of nature.#

  • Light is a wave and a particle. An electron is also a particle and a wave. Is everything a wave and a particle? The answer is YES! This is what is meant by wave-particle duality. Sometimes we only see one side of the duality because under some conditions, either wave or particle characteristics are more pronounced.

  • The wave-like and particle-like characteristics of a physical entity are inversely proportional to each other as described by the de Broglie relationship.

De Broglie comes up with a simple equation which establishes the duality of matter.#

\[\boxed{\lambda = \frac{h}{p}}\]

  • Where \(h\): Planck’s constant. \(p\): the momentum of the object (electron, photon, molecule, chair, etc.). \(\lambda\): wavelength associated with the object.

  • The relation implies that heavy objects have a small wavelength, and light objects have a large wavelength. Hence the smaller the object, the more pronounced wave-like qualities it will have. And vice versa, the bigger the object, the more particle-like qualities it will have.